Search Results (7)

Recently, a new type of derivative has been introduced, known as Caputo proportional derivatives. These are motivated by the applications of such derivatives (which are a generalization of Caputo’s standard fractional derivative) and the need to incorporate such calculus into the research on operators. The investigation therefore focuses on the equivalence of differential and integral problems for proportional calculus problems. The operators are always studied in the appropriate function spaces. Furthermore, the investigation extends these results to encompass the more general notion of Hilfer hybrid derivatives. The primary aim of this study is to preserve the maximal regularity of solutions for this class of problems. To this end, we consider such operators not only in spaces of absolutely continuous functions, but also in particular in little Hölder spaces. It is widely acknowledged that these spaces offer a natural framework for the study of classical Riemann–Liouville integral operators as inverse operators with derivatives of fractional order. This paper presents a comprehensive study of this problem for proportional derivatives and demonstrates the application of the obtained results to Langevin-type boundary problems. Full article

Inspired by prior research on fractional calculus, we introduce new fractional integral and derivative operators: the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral and the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional derivative. Numerous previous studied fractional integrals and derivatives can be considered as particular instances of the novel operators introduced above. Some properties of the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral are discussed, including mapping properties, the generalized Laplace transform of the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral and $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional derivative. The results obtained suggest that the most comprehensive formulation of this fractional calculus has been achieved. Under the guidance of the findings from earlier sections, we investigate the existence of mild solutions for the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional Cauchy problem. An illustrative example is provided to demonstrate the main results. Full article

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the $p$-proportional $\omega$-weighted $\kappa$-Hilfer derivative includes an exponential function, $D_{a,\lambda }^{\sigma ,\rho ,p,\kappa ,\omega }$, and then we consider a non-instantaneous impulse differential inclusion containing $D_{a,\lambda }^{\sigma ,\rho ,p,\kappa ,\omega }$ with order $\sigma \in (1,2)$ and of kind $\rho \in [0,1]$ in Banach spaces. We deduce the relevant relationship between any solution to the studied problem and the integral equation that corresponds to it, and then, by using an appropriate fixed-point theorem for multi-valued functions, we give two results for the existence of these solutions. In the first result, we show the compactness of the solution set. Next, we introduce the concept of the $(p,\omega ,\kappa )$-generalized Ulam-Hyeres stability of solutions, and, using the properties of the multi-valued weakly Picard operator, we present a result regarding the $(p,\omega ,\kappa )$-generalized Ulam-Rassias stability of the objective problem. Since many fractional differential operators are particular cases of the operator $D_{a,\lambda }^{\sigma ,\rho ,p,\kappa ,\omega }$, our work generalizes a number of recent findings. In addition, there are no past works on this kind of fractional differential inclusion, so this work is original and enjoyable. In the last section, we present examples to support our findings. Full article

Despite the existence of a secure and reliable immunization, measles, also known as rubeola, continues to be a leading cause of fatalities globally, especially in underdeveloped nations. For investigation and observation of the dynamical transmission of the disease with the influence of vaccination, we proposed a novel fractional order measles model with a constant proportional (CP) Caputo operator. We analysed the proposed model’s positivity, boundedness, well-posedness, and biological viability. Reproductive and strength numbers were also verified to examine how the illness dynamically behaves in society. For local and global stability analysis, we introduced the Lyapunov function with first and second derivatives. In order to evaluate the fractional integral operator, we used different techniques to invert the PC and CPC operators. We also used our suggested model’s fractional differential equations to derive the eigenfunctions of the CPC operator. There is a detailed discussion of additional analysis on the CPC and Hilfer generalised proportional operators. Employing the Laplace with the Adomian decomposition technique, we simulated a system of fractional differential equations numerically. Finally, numerical results and simulations were derived with the proposed measles model. The intricate and vital study of systems with symmetry is one of the many applications of contemporary fractional mathematical control. A strong tool that makes it possible to create numerical answers to a given fractional differential equation methodically is symmetry analysis. It is discovered that the proposed fractional order model provides a more realistic way of understanding the dynamics of a measles epidemic. Full article

In this paper, we establish existence and uniqueness results for a new class of boundary value problems involving the $\psi$-Hilfer generalized proportional fractional derivative operator, supplemented with mixed nonlocal boundary conditions including multipoint, fractional integral multiorder and derivative multiorder operators. The given problem is first converted into an equivalent fixed point problem, which is then solved by means of the standard fixed point theorems. The Banach contraction mapping principle is used to establish the existence of a unique solution, while the Krasnosel’skiĭ and Schaefer fixed point theorems as well as the Leray–Schauder nonlinear alternative are applied for obtaining the existence results. We also discuss the multivalued analogue of the problem at hand. The existence results for convex- and nonconvex-valued multifunctions are respectively proved by means of the Leray–Schauder nonlinear alternative for multivalued maps and Covitz–Nadler’s fixed point theorem for contractive multivalued maps. Numerical examples illustrating the obtained results are also presented. Full article

This paper studies the existence and uniqueness of solutions for a coupled system of Hilfer-type generalized proportional fractional differential equations supplemented with nonlocal asymmetric multipoint boundary conditions. We consider both the scalar and the Banach space case. We apply standard fixed-point theorems to derive the desired results. In the scalar case, we apply Banach’s fixed-point theorem, the Leray–Schauder alternative, and Krasnosel’skiĭ’s fixed-point theorem. The Banach space case is based on Mönch’s fixed-point theorem and the technique of the measure of noncompactness. Examples illustrating the main results are presented. Symmetric distance between itself and its derivative can be investigated by replacing the proportional number equal to one half. Full article

In this paper, we discuss the existence and uniqueness of solutions for boundary value problems for Hilfer generalized proportional fractional differential equations with multi-point boundary conditions. Firstly, we consider the scalar case for which the uniqueness result is proved by using Banach’s fixed point theorem and the existence results are established via Krasnosel’skiĭ’s fixed point theorem and Leray–Schauder nonlinear alternative. We then establish an existence result in the Banach space case based on Mönch’s fixed point theorem and the technique of the measure of noncompactness. Examples are constructed to illustrate the application of the main results. We emphasize that, in this paper, we initiate the study of Hilfer generalized proportional fractional boundary value problems of order in (1, 2]. Full article